Hyperbolic paraboloid: a cardboard model

The card­board model of the hyper­bolic parab­o­loid is amazing. Its name includes words that are only studied in high school (sepa­rately, but together only at univer­sity). It allows one to see clearly the non-trivial prop­er­ties of this complex surface. It is also possible to make and discuss such a model even with chil­dren.

One can see an animated defi­n­i­tion of the hyper­bolic parab­o­loid and under­stand where there are parabolas and where there are hyper­bolas by watching the film Saddle surface: hyper­bolic parab­o­loid. And the prop­erty of the linearity of this saddle surface — that it is formed by the motion of a straight line — can be demon­strated by doing the coun­ter­in­tu­itive chip exper­i­ment shown in Chips: hyper­bolic parab­o­loid.

The card­board model allows one to see even more clearly the linearity prop­erty of the hyper­bolic parab­o­loid — it explic­itly presents gener­a­trix from both fami­lies. Again clearly the card­board model also allows to make sure that the surface of the hyper­bolic parab­o­loid is “not flat” (has nega­tive Gaussian curva­ture) — a piece of paper cannot be placed on it without crum­pling.

The model of hyper­bolic parab­o­loid under consid­er­a­tion is bend­able: it can be shrunk by keeping the pieces of paper parallel in each of the two fami­lies, but by changing the angle between these fami­lies. The inter­sec­tion points of the forma­tions will obvi­ously remain unchanged, the surface itself will change, but will always remain a hyper­bolic parab­o­loid. The right angle case corre­sponds to the “school” rectan­gular hyper­bolas when their asymp­totes are perpen­dic­ular. Theo­ret­i­cally, you could make such a panorama book (Pop Up), but if you use unbend­able (respec­tively, rela­tively thick) card­board, the number of layers is too high for it to close “flat” well.